《金融经济学原理》课程教学资源（文献资料）When uncertainty and volatility are disconnected - Implications for asset pricing and portfolio performance.pdf_大学文库

Journal of Econometrics xxx (xxxx) xxx Please cite this article as: Yacine Aït-Sahalia et al., Journal of Econometrics, https://doi.org/10.1016/j.jeconom.2023.105654 0304-4076/© 2024 Elsevier B.V. All rights reserved. Contents lists available at ScienceDirect Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom When uncertainty and volatility are disconnected: Implications for asset pricing and portfolio performance✩ Yacine Aït-Sahalia a,b,∗ , Felix Matthys c , Emilio Osambela d , Ronnie Sircar e a Department of Economics and Bendheim Center for Finance, Princeton University, United States of America b NBER, United States of America c Instituto Tecnológico Autónomo de México (ITAM), Mexico d Board of Governors of the Federal Reserve System, United States of America e Department of Operations Research & Financial Engineering and Bendheim Center for Finance, Princeton University, United States of America A R T I C L E I N F O JEL classification: G11 G12 G17 Keywords: Risk Uncertainty Volatility Robust control Portfolio choice Asset returns Equity risk premium A B S T R A C T We analyze an environment where the uncertainty in the equity market return and its volatility are both stochastic and may be potentially disconnected. We solve a representative investor’s optimal asset allocation and derive the resulting conditional equity premium and risk-free rate in equilibrium. Our empirical analysis shows that the equity premium appears to be earned for facing uncertainty, especially high uncertainty that is disconnected from lower volatility, rather than for facing volatility as traditionally assumed. Incorporating the possibility of a disconnect between volatility and uncertainty significantly improves portfolio performance, over and above the performance obtained by conditioning on volatility only. ‘‘You would think if uncertainty was high, you’d have a bit more volatility.’’ [William Dudley, New York Fed President, February 15, 2017.] 1. Introduction Although the notions of uncertainty and volatility are often used interchangeably, the two concepts are inherently different: volatility measures the dispersion of short-term shocks around a long-term mean, while uncertainty measures the difficulty to forecast the distribution of returns, including its long-term mean. Fig. 1 contains a scatter plot of volatility and uncertainty at the weekly frequency between January 1986 and December 2020, proxied respectively by realized volatility computed from high-frequency ✩ For valuable comments and suggestions, we thank the Editor, an Associate Editor and two referees, as well as Richard H. Clarida, Andrew Detzel, Ian Dew-Becker, William H. Janeway, Pascal Maenhout, Cameron Peng, Valery Polkovnichenko, Yan Qian, Gustavo A. Suarez and Laura Veldkamp. We are also grateful for comments from participants at presentations given at the University of Zurich, ETH Zurich, the Board of Governors of the Federal Reserve System, Erasmus University Rotterdam, the FRB-NYU workshop on risk and uncertainty, the International Association for Applied Econometrics, the 2021 SIAM Annual Meeting, the Paris December Finance Meetings and the SFS Cavalcade North America 2022. The views in this paper do not necessarily reflect those of the Federal Reserve System or its Board of Governors. ∗ Corresponding author at: Department of Economics and Bendheim Center for Finance, Princeton University, United States of America. E-mail addresses: yacine@princeton.edu (Y. Aït-Sahalia), felix.matthys@itam.mx (F. Matthys), emilio.osambela@frb.gov (E. Osambela), sircar@princeton.edu (R. Sircar). https://doi.org/10.1016/j.jeconom.2023.105654 Received 28 December 2022; Received in revised form 28 July 2023; Accepted 10 November 2023

Journal of Econometrics xxx (xxxx) xxx 2 Y. Aït-Sahalia et al. Fig. 1. Uncertainty and volatility regimes. Notes: The figure shows a scatter plot of standardized uncertainty (proxied by the economic policy uncertainty index EPU𝑡 ) and volatility (proxied by realized volatility). Both are sampled at the weekly frequency from January 1986 until December 2020. The threshold values for volatility and uncertainty are given by their mean plus one half of their standard deviation. We say that volatility and uncertainty are high (respectively, low), when they are above (respectively, below) their threshold values. High disconnect occurs when either uncertainty is high while volatility is low (denoted ‘‘HL’’) or when uncertainty is low while volatility is high (denoted ‘‘LH’’). In the other two quadrants, uncertainty and volatility are in sync and disconnect is low. data and the economic policy uncertainty index (EPU) of Baker et al. (2016). The figure shows that the two variables, although generally positively correlated, are far from being perfect substitutes for one another. The degree of connection between uncertainty and volatility appears to vary across periods. It is natural a priori to expect uncertainty and volatility to be in general positively correlated, as the quote above implies. For example, the theoretical model of Pastor and Veronesi (2013) implies such a positive relationship;1 while Amengual and Xiu (2018) show that the resolution of monetary policy uncertainty is generally associated with declines in volatility. Yet, there have been several episodes in which either volatility was high and uncertainty was significantly lower or vice versa. For instance, the US 2016 presidential election generated some uncertainty about long-term economic and other policies, but was surprisingly characterized by very low levels of stock market volatility. Similarly, the UK’s exit from the EU (Brexit) involved substantial uncertainty about trade, growth, and immigration policies for the UK and the EU, but had a barely noticeable impact on short-term volatility in their respective stock markets.2 These events are examples of situations in which a disconnect between the two variables appeared because uncertainty was substantially higher than volatility. By contrast, the stock market dynamics during the financial crisis in 2008 and the initial stock market reaction to the diffusion of the Covid-19 pandemic in the spring of 2020 are examples of situations in which a disconnect occurred due to a higher increase in volatility compared to the rise in uncertainty. Interestingly, in the months following the stock market crash in March 2020, volatility declined much faster than uncertainty, leading to a switch in the nature of disconnect, characterized instead by uncertainty being higher than volatility. Separately, the empirical evidence regarding the risk-return trade-off, with risk measured by volatility, is mixed: while many papers find a positive relationship,3 as predicted by standard theory, equally many find a negative one,4 depending on the sample period and methodology, whether total or idiosyncratic volatility is considered, and other distinctions.5 This suggests that a model where investors worry only about volatility is perhaps too parsimonious. Motivated by these empirical observations, we propose a model in which uncertainty and volatility are two separate stochastic processes, whose degree of connection is stochastic. The representative investor in our model can be averse to both volatility and uncertainty, with possibly different degrees of aversion to each. Going back to the simple regimes exhibited in Fig. 1, when we solve the model, we obtain a different equity premium, risk-free rate, and portfolio strategy in regimes in which uncertainty and volatility are high or low at the same time (connected), compared to regimes in which one of them is significantly higher than the 1 At least if the precision of political signals in their model is constant over time. If it is allowed to vary then it is possible for the relationship to be reversed. 2 See Bialkowski et al. (2022) for a related discussion. 3 See, e.g., Harvey (1989), Campbell and Hentschel (1992), Harrison and Zhang (1999), Ghysels et al. (2005) and Ludvigson and Ng (2007). 4 See, e.g., Campbell (1987), Breen et al. (1989), Nelson (1991), Glosten et al. (1993) and Brandt and Kang (2004). See also (Blitz et al., 2020) for an overview of the low-risk anomaly, including the empirical finding of a negative risk-return trade off and a discussion of potential theoretical explanations for this empirical result. 5 See, e.g., Merton (1980), French et al. (1987), Goyal and Santa-Clara (2003), Ang et al. (2009) and Campbell et al. (2018)

Journal of Econometrics xxx (xxxx) xxx 3 Y. Aït-Sahalia et al. other (disconnected). We show that incorporating this potential disconnect leads to substantially improved forecasts of the equity premium and portfolio performance. To allow for a distinction between uncertainty and volatility, the decision maker should be uncertain about the correct probability model to use. To model this, we rely on the robust control approach pioneered by Hansen and Sargent (2001).6 The representative investor recognizes that he or she is unable to know exactly the true underlying model, and considers instead a set of models that are statistically difficult to distinguish from one another, seeking consumption and investment policies that perform well across that full set of models. Uncertainty measures the radius of the set of potential models: when uncertainty is high, the range of potential models around the true model expands. How much the investor responds to this expansion is controlled by a parameter, which we think of as the investor’s coefficient of uncertainty aversion. We then represent uncertainty in the form of the product of stochastic volatility and a ‘‘disconnect’’ stochastic process, that drives a wedge between volatility and uncertainty. An investor interested in policies that are robust across the set of alternative models optimally evaluates his or her policies under the worst-case alternative in the set of models under consideration. For that reason, the dynamics of volatility and disconnect endogenously have an impact on asset prices and the associated optimal policies of the investor. We first solve the model in partial equilibrium, computing in closed form a representative log-utility investor’s optimal policies for consumption and portfolio choice. We show that the investor’s optimal equity holding consists of the standard myopic term, which is inversely proportional to stock return variance, and a new (yet still myopic as befits a logarithmic investor) uncertainty correction term, which is the optimal response to the potential disconnect between volatility and uncertainty. We find that the contemporaneous interaction of volatility and uncertainty plays an important role in determining optimal portfolios. In particular, the sensitivity of portfolio weights to changes in volatility depends on the level of model uncertainty. Accordingly, the trajectory of portfolio weights in a given period depends on the joint dynamics of volatility and uncertainty, including whether they are connected or disconnected. Given the optimal asset allocation by a representative investor, we solve for the equilibrium equity premium and risk-free rate. We obtain explicit formulae, which are nonlinear functions of both stochastic volatility and disconnect. They predict that the uncertainty term embedded in disconnect generates a flight-to-quality-like correlation among asset returns. In high-uncertainty periods, investors require a high equity premium to hold the risky asset and are willing to accept a low risk-free rate to hold the safe asset. The presence of stochastic volatility may amplify or diminish these effects depending on whether it is connected or disconnected from uncertainty, respectively. The interaction of volatility with a possibly disconnected uncertainty means that our model can generate a high equity premium in a low volatility environment, whenever the disconnect with uncertainty is high. These empirical patterns were observed in the period surrounding the US 2016 election, among other disconnected episodes. Our model’s results can help explain the challenges faced by previous empirical studies trying to establish a risk-return trade-off using volatility alone as a measure of risk. As noted above, while most theoretical asset pricing models imply a positive relationship between return and risk, the empirical evidence for such a trade-off is in reality mixed or inconclusive. Although our model also implies a positive relationship between return and volatility, consistent with the assumption of traditional risk (i.e., volatility) aversion, it adds a new component in the equity premium associated with the disconnect between volatility and uncertainty, which is theoretically positive. Including this component allows our model to more accurately reproduce asset pricing patterns. We find that uncertainty is a strong positive predictor of the equity premium, so even in periods when the traditional volatility-return trade off fails to materialize, or has the wrong sign, the predicted equity premium in our model is increasing in uncertainty due to the uncertainty-return trade off we find. Our results show that allowing for uncertainty to be disconnected from volatility makes uncertainty a much better variable than volatility in terms of generating a trade-off with expected returns: it appears from our results that the equity premium is earned mainly for facing uncertainty, especially high uncertainty that is disconnected from lower volatility, rather than for facing volatility per se.7 We then go on to evaluate the practical value of the equity premium formula that emerges from the model, by examining the portfolio performance of a reference investor who predicts future excess returns using our estimated relation between stock excess return, volatility, and uncertainty.8 We find that our model significantly improves portfolio performance relative to both existing 6 See also (Anderson et al., 2003), Hansen et al. (2006), Hansen and Sargent (2008), and Hansen and Sargent (2011). An axiomatic justification for the approach is provided by Strzalecki (2011). Alternative frameworks can be employed to capture robust decision making. Ambiguity aversion is one such formulation: agents there prefer options whose probabilities are known with certainty over those whose probabilities are uncertain. Info-gap models of uncertainty are another approach: there, decision-makers choose actions that minimize the maximum possible regret, where regret is the difference between the payoff obtained and the payoff that would have been obtained had the decision-maker known the true probability distribution. In the maxmin expected utility approach, decision-makers maximize their expected utility using the worst-case probability distribution, rather than a specific probability distribution that is known to be correct; see Gilboa and Schmeidler (1989), and Kochov (2015) for an axiomatic justification. For a discussion and comparison of maxmin expected utility and robust control, see Hansen and Sargent (2001), Hansen et al. (2006) and Chen and Epstein (2002). 7 The relation between uncertainty and returns has been considered at least since Keynes’ General Theory (Keynes, 1936, p. 148): “It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain. It is reasonable, therefore, to be guided to a considerable degree by the facts about which we feel somewhat confident, even though they may be less decisively relevant to the issue than other facts about which our knowledge is vague and scanty. (. . ) The state of long-term expectation, upon which our decisions are based, does not solely depend, therefore, on the most probable forecast we can make. It also depends on the confidence with which we make this forecast—on how highly we rate the likelihood of our best forecast turning out quite wrong.” We provide empirical validation for this relation, in a setup in which volatility also drives returns. 8 We deliberately keep our empirical analysis simple given the fact that econometric inference of non-linear conditional asset pricing model with unknown parameters, which over model belongs to, is challenging due to miss specification or weak identification. A general discussion of this issue can be found in Antoine et al. (2020)

Journal of Econometrics xxx (xxxx) xxx 4 Y. Aït-Sahalia et al. unconditional and conditional asset pricing models, including those that time volatility but do not account for its potential disconnect from uncertainty, and those that allow for the presence of uncertainty but not for its disconnect from volatility. These results are valid when back-testing the model in-sample but also out-of-sample. The main contribution of the paper is therefore the construction of a simple and intuitive model where a representative investor can be averse to potentially disconnected volatility and uncertainty, which can be solved explicitly and results in equilibrium formulae for the equity premium and risk-free rate that perform well empirically. The rest of the paper is organized as follows. We start with a brief discussion of the related literature in Section 2. Section 3 sets up the model, in which volatility and uncertainty are potentially disconnected. Section 4 solves for the optimal consumption and portfolio allocation taking prices as given. Section 5 solves for the equilibrium asset prices’ dynamics, including the conditional equity premium and risk-free rate. Section 6 describes the data and our empirical analysis. Section 7 evaluates the performance of our model’s implied portfolio strategy in the full sample and in specific high-disconnect regimes. Section 8 shows that our results also hold out-of-sample. Section 9 concludes. The Appendix contains technical material and proofs of the propositions in the paper. 2. Related literature Our work is related to a growing literature that analyzes the effect of uncertainty on asset prices. While we do not model the source of uncertainty, different approaches can be employed to justify the presence of a disconnect between uncertainty and volatility. The model of Pastor and Veronesi (2012) and Pastor and Veronesi (2013) implies that political uncertainty shocks command a risk premium and stocks become more correlated and volatile in periods of elevated political uncertainty. If the constant signal precision in their model was generalized to be stochastic, a disconnect would arise. Alternatively, Barroso and Detzel (2021) show that the performance of the volatility-managed portfolios in Moreira and Muir (2017) is increasing in a sentiment variable, consistent with prior theory by citeyuyuan11 showing that sentiment traders are expected to under-react to volatility. A stochastic sentiment can be a source of disconnect as well. Bidder and Dew-Becker (2016) show that a model with recursive preferences and uncertainty about the dynamics of consumption is consistent with a large and time-varying equity premium. On the empirical side, Brogaard and Detzel (2015) show that uncertainty positively forecasts excess returns and innovations in uncertainty carry a significantly negative risk premium, while Bali et al. (2017) find that the difference between returns on portfolios with the highest and lowest uncertainty beta is negative and highly significant. In our paper, the fact that the equity premium responds to both volatility and uncertainty is not assumed, however. It occurs endogenously in the model in equilibrium because the representative investor is averse to volatility (risk-averse in the standard sense) but also averse to uncertainty (by seeking robustness to a range of potential models that may generate the observed empirical realizations). In existing models that do not account for model uncertainty, time-varying volatility drives the optimal policies of a risk-averse agent, but there is no mechanism to account for the uncertainty attached to the assumptions of the model (see, for example, Chacko and Viceira (2005), Liu (2007), and Drechsler and Yaron (2011)). On the other hand, in models in which the agent is averse to model uncertainty but volatility is constant, the optimal portfolio strategy is to reduce the exposure to risky assets when uncertainty increases (see, for example, Trojani and Vanini (2000), Maenhout (2004), Maenhout (2006), and Illeditsch (2011)). Since volatility and returns tend to be negatively correlated, the resulting conservative asset allocation makes the investor forgo much of the upside of asset markets in situations in which volatility fails to materialize despite levels of uncertainty above or close to average, such as the aftermaths of the US 2016 election, Brexit, and the months following the stock market crash of March 2020 driven by the pandemic. We show that including uncertainty and volatility separately allows the investor to take advantage of such situations. The volatility-managed portfolios in Moreira and Muir (2017), which take less risk when volatility is high, produce large alphas and Sharpe ratios. We show that incorporating disconnect as an additional variable achieves even higher portfolio performance. Liu et al. (2005) consider a setup with constant diffusive volatility but in which the stock price is subject to jump risk and the investor is averse to uncertainty with respect to jumps. They show that their model is consistent with several empirical patterns of option prices. Jahan-Parvar and Liu (2014) show that a production-based asset-pricing model with regime-switching productivity, constant volatility in each regime and ambiguity aversion can reproduce, among other empirical patterns, predictability of excess return by investment-capital, price-dividend, and consumption-wealth ratios.9 In contrast to these papers, in our model both stochastic volatility and uncertainty predict excess returns, and we show empirically that this predictability generates high portfolio performance, especially in periods of high uncertainty that is disconnected from low volatility. Similar to our work, other papers have investigated the differential impact of volatility and uncertainty on equity returns. In particular, in line with the results found in our paper, Anderson et al. (2009) find stronger empirical evidence for an uncertainty - return trade-off rather than for the traditional risk-return trade-off.10 Bekaert et al. (2009) find that the equity risk premium is primarily driven by time-varying risk aversion, while uncertainty is the main driver of counter-cyclical volatility of asset returns, suggesting that uncertainty affects second rather than first return moments.11 In contrast to these papers, however, we device a 9 Gallant et al. (2019) show that in several model specifications, including the one used by Jahan-Parvar and Liu (2014), having model uncertainty as a feature of the model provides a better empirical asset pricing fit. 10 While Anderson et al. (2009) also employ a robust control setup similar to ours to model time varying uncertainty and volatility, they make use of different empirical proxies for both uncertainty and volatility: First, their measure of uncertainty is based on forecaster’s disagreement, while we use the economic policy uncertainty (EPU) index as our empirical measure for uncertainty. Second, we use a non-parametric estimator for our measure of conditional volatility, while they construct their conditional volatility proxy via a Mixed Data Sampling, or MIDAS, approach. 11 In their model, uncertainty is proportional to the conditional volatility of dividend growth, while time-varying risk aversion is introduced via an external habit stock in the utility function of the representative investor

Journal of Econometrics xxx (xxxx) xxx 5 Y. Aït-Sahalia et al. dynamic portfolio strategy that not only exploits the fact that uncertainty and volatility have a differential impact on equity returns, but simultaneously takes advantage of the different time series properties of volatility and uncertainty. The theoretical framework in this paper is related to the one presented in Sbuelz and Trojani (2008), which also considers a general equilibrium model with an ambiguity averse agent and stochastically varying uncertainty. However, while in their framework the uncertainty or ambiguity set is constrained by the stochastic state of the economy, we allow for a more flexible specification in which time-varying uncertainty is driven by both stochastic volatility and a disconnect process. Drechsler (2013) considers a general framework with stochastic model uncertainty, stochastic volatility and jumps. While his focus is on reproducing the empirical properties of index options and the variance premium, ours is on stock excess return predictability and the characterization of asset prices, especially by focusing on the importance of periods of high disconnect between uncertainty and volatility. Faria and Correia-da Silva (2016) study optimal asset allocation for an investor subject to stock return stochastic volatility and constant ambiguity uncertainty. Like us, they are interested in the impact of model uncertainty on optimal portfolios in a setup with stochastic volatility. However, because they assume constant model uncertainty, their setup is not suited to study the joint dynamics of volatility, uncertainty, and their disconnect on the optimal portfolio decision. Moreover, they do not study asset prices in equilibrium and their main focus is on the marginal impact of model uncertainty on investors’ hedging demands, which they find to be very low. Brenner and Izhakian (2018) study the joint impact of risk and (ambiguity) uncertainty on the equity premium. While the research question they ask is related to ours, they consider a very different framework. First, their measure of uncertainty is independent of risk by construction. Accordingly, their setup would not allow to study time-varying levels of disconnect between volatility and uncertainty, which is central in our paper. Second, while we let the representative investor be averse to both risk and uncertainty, Brenner and Izhakian (2018) allow for the investor to be ambiguity loving in some states of nature. Lastly, while in our framework uncertainty and volatility are exogenous, a theoretical foundation for the disconnect process may be obtained in models in which the representative agent learns about the true model driving the economy.12 3. A framework with volatility and uncertainty possibly disconnected We consider an infinite-horizon expected-utility maximization problem where a representative investor chooses his or her consumption level as well as allocates his or her funds between a risk-less and a risky asset, which exhibits stochastic volatility. For this purpose, the investor employs a benchmark or reference model that represents his or her best estimate of the risky asset’s dynamics. However, the investor fears that the model he or she uses is potentially incorrect, and worries that the true model could lie in a set of alternative models that are statistically difficult to distinguish from the reference model. To cope with model uncertainty, the investor chooses optimal consumption and portfolio holdings that are robust across the set of alternative models. The size of the set of alternative models is our proxy for how much uncertainty the representative investor faces. When the radius of the set is large, alternatives that are statistically far from the reference model will be considered, and the investor will face high uncertainty about the reference model and vice versa. We generalize the radius of the set of alternative models relative to the classical robust control literature. We first let stochastic volatility affect it, so that model uncertainty is ceteris paribus higher when there is higher volatility and vice versa. To avoid perfect correlation between model uncertainty and volatility, however, we introduce an additional stochastic process which drives the degree of connection between uncertainty and volatility. This setup allows us to study market scenarios in which, simultaneously, uncertainty can be potentially high while volatility is low and vice versa, such as the ‘‘HL’’ and ‘‘LH’’ regimes characterized in Fig. 1. Furthermore, we allow the investor to react differently to the same amount of uncertainty, through a coefficient of uncertainty aversion. An investor with a higher degree of uncertainty aversion would seek robustness against a larger set of models than an investor with a lower one, for the same amount of overall uncertainty. In the rest of this section, we formalize this modeling framework. 3.1. From the reference to the robust model Assume a complete, filtered probability space (𝛺, , P) satisfying the usual assumptions, where P denotes the reference probability measure that represents the investor’s best estimate of the risky-asset dynamics. We consider a Lucas-tree economy (see Lucas, 1978) with a single risky asset with price 𝑆𝑡 and a risk-free asset with price 𝐵𝑡 . The risky asset is a perpetual claim on the stream of aggregate dividends 𝐷𝑡 . The dynamics of dividends and asset prices under the reference probability measure are 𝑑𝐷𝑡 𝐷𝑡 = 𝜇𝐷𝑑𝑡 + 𝜎𝐷𝑣𝑡𝑑𝑊 𝐷 𝑡 , 𝐷0 > 0, (1) 𝑑𝐵𝑡 𝐵𝑡 = 𝑟𝑓,𝑡𝑑𝑡, 𝐵0 > 0, (2) 𝑑𝑆𝑡 𝑆𝑡 = ( 𝜇𝑆,𝑡 − 𝐷𝑡 𝑆𝑡 ) 𝑑𝑡 + 𝜎𝑆𝑣𝑡𝑑𝑊 𝑆 𝑡 , 𝑆0 > 0, (3) 12 See for instance Epstein and Schneider (2007), Leippold et al. (2007) and Epstein and Schneider (2008), for frameworks in which the agent learns about the fundamentals of the economy under ambiguity and the quality of the information may be high or low. In Dew-Becker and Nathanson (2019), agents trying to learn about the fundamentals of the economy might attenuate the most severe effects of uncertainty (or ambiguity). Thus, translated to our setup, an agent who acquires information about the true dynamics of the stock price process may reduce the uncertainty about its fundamentals provided that the quality of information is sufficiently high and therefore affects the degree of disconnect between uncertainty and volatility

Journal of Econometrics xxx (xxxx) xxx 6 Y. Aït-Sahalia et al. 𝑑𝑣𝑡 = 𝜇𝑣,𝑡𝑑𝑡 + 𝜎𝑣,𝑡 (√ 1 − 𝜌 2 𝑆,𝑣𝑑𝑊 𝑣 𝑡 + 𝜌𝑆,𝑣𝑑𝑊 𝑆 𝑡 ) , 𝑣0 > 0, (4) where 𝑊 𝐷 𝑡 , 𝑊 𝑆 𝑡 and 𝑊 𝑣 𝑡 are Brownian motions, 𝜇𝐷 and 𝜎𝐷 are the drift and volatility of dividend growth, 𝑟𝑓,𝑡 is the risk-free rate, 𝜇𝑆,𝑡 is the expected total return of the risky asset and 𝜎𝑆 is a constant scaling stock return volatility.13 The stochastic volatility of stock return and dividends growth, 𝑣𝑡 , is a general stochastic process with drift 𝜇𝑣,𝑡 and diffusion 𝜎𝑣,𝑡 under the reference probability measure.14 The term 𝜌𝑆,𝑣 captures the correlation between the stock return and its volatility. In order to formally introduce model uncertainty, we define by P 𝜗 the robust probability measure, where 𝜗𝑡 is the change from the reference measure P to the robust measure P 𝜗 . This implies that a robust investor considers alternative models of the form:15 𝑑𝑆𝑡 𝑆𝑡 = ( 𝜇𝑆,𝑡 − 𝐷𝑡 𝑆𝑡 + 𝜎𝑆𝑣𝑡ℎ𝑡 ) 𝑑𝑡 + 𝜎𝑆𝑣𝑡𝑑𝑊 𝑆,𝜗 𝑡 , (5) where 𝑊 𝑆,𝜗 𝑡 is also a Brownian motion, but now under the robust probability measure P 𝜗 . Importantly, the drift of the stock return is now perturbed by the new term 𝜎𝑆𝑣𝑡ℎ𝑡 , driven by both stochastic volatility 𝑣𝑡 and the drift perturbation function ℎ𝑡 which will be optimally chosen by the representative agent.16 As can be seen from Eq. (5), the higher the perturbation function ℎ𝑡 becomes, the larger becomes the drift distortion of the risky asset and hence, the higher is the investor’s demand for robustness against potential model miss-specification. Conversely, if ℎ𝑡 = 0, then the representative investor has full confidence in his or her reference model. 3.2. The size of the alternative set of models and the role of disconnect While a robust investor considers a variety of alternative models, not all of them are plausible, i.e. they may be too distinct to be considered as reasonably close to the reference model. To discipline the size of the alternative model set, we make use of relative entropy which is a convenient measure of the distance between the reference model and the alternative models. The growth in entropy of P 𝜗 relative to P over the time interval [𝑡, 𝑡 + 𝛥𝑡] is defined as 𝐺(𝑡, 𝑡 + 𝛥𝑡) ≡ E 𝜗 𝑡 [ log ( 𝜗𝑡+𝛥𝑡 𝜗𝑡 )] , (8) and the instantaneous growth rate of relative entropy at time 𝑡 is then given by (𝜗𝑡 ) ≡ lim 𝛥𝑡→0 𝐺(𝑡, 𝑡 + 𝛥𝑡) 𝛥𝑡 = 1 2 ℎ 2 𝑡 , (9) where the last equality is proven in Appendix A.1. When ℎ𝑡 = 0, the relative entropy growth rate is zero, which implies that the two probability measures are equivalent. As ℎ𝑡 increases, so does the distance between the reference model and the alternative models. Next, we restrict the set of alternative models under consideration by the investor in the form of an upper bound on this distance. We assume that set of admissible alternative models is bounded from above as follows { 𝜗𝑡 ∶ (𝜗𝑡 ) ≤ 𝜖 2 2  2 𝑡 } , (10) where 𝑡 denotes the stochastic model uncertainty and the constant parameter 𝜖 measures the investor’s degree of uncertainty aversion. When 𝜖 = 0 the set of alternative models is empty and the investor has full confidence in the reference model. By contrast, an investor with higher 𝜖 expands the set of alternative models to include models that are statistically further away from the reference model. Finally, in order to formally introduce disconnect, we posit the following functional form for modeling uncertainty 𝑡 = 𝜂𝑡𝑣𝑡 . (11) 13 We will show, that in equilibrium, the stock price is proportional to dividends, so that 𝑊 𝐷 𝑡 = 𝑊 𝑆 𝑡 , ∀𝑡 ≥ 0. Therefore, instead of specifying the correlation between dividends 𝐷𝑡 and our state variables in the model, we specify the correlations with respect to the stock price process directly. 14 Given logarithmic preferences, the solution is independent of the specific choice of a drift and diffusion process for volatility. The investor is myopic with regards to the dynamics of volatility: he or she cares only about the current value of the state variables. However, the drift and volatility of volatility, 𝜇𝑣,𝑡 and 𝜎𝑡,𝑣, are not fully unrestricted since we require the volatility process to remain stationary and positive. Specific restrictions to be imposed are model-dependent. As an example, Feller’s square-root process in which the drift term is linearly mean reverting, i.e. 𝜇𝑣,𝑡 ∶= 𝜅 ( 𝜃 − 𝑣𝑡 ) with 𝜃 > 0 and 𝜅 > 0 and the volatility is given by 𝜎𝑣,𝑡 ∶= 𝜎 √ 𝑣𝑡 , 𝜎 > 0 requires that the parameters satisfy 𝜅𝜃 > 𝜎 2 2 for the process 𝑣𝑡 to be precluded from reaching zero. 15 A similar perturbed equation for the dividend process is omitted for brevity. Since it turns out that the stock price will be proportional to dividends in equilibrium, adding that equation here is not necessary. 16 This result follows immediately from an applications of Girsanov’s theorem which states that for any 𝑡 -measurable random variable 𝑍 and 𝑇 > 𝑡 we can write E 𝜗 [ 𝑍𝑇 | | 𝑡 ] = E [ 𝜗𝑇 𝜗𝑡 𝑍𝑇 | | | | 𝑡 ] . (6) so that the Brownian Motions are related by 𝑑𝑊 𝑆 𝑡 = 𝑑𝑊 𝑆,𝜗 𝑡 + ℎ𝑡𝑑𝑡 where 𝜗𝑡 is an exponential martingale, with dynamics 𝑑𝜗𝑡 𝜗𝑡 = ℎ𝑡𝑑𝑊 𝑆 𝑡 . (7)

Journal of Econometrics xxx (xxxx) xxx 7 Y. Aït-Sahalia et al. where we refer to 𝜂𝑡 as our stochastic disconnect process.17 This specification is motivated by the empirical relation between volatility and uncertainty observed in Fig. 1, showing that while volatility and uncertainty are in general positively correlated, there are situations in which one of them is significantly higher than the other, i.e., they are disconnected. The stochastic process 𝜂𝑡 measures the degree of disconnect, is positive and normalized to have mean one. This normalization is for ease of interpretation of the level of disconnect relative to its mean: when 𝜂𝑡 is far away from one (either above or below) there is high disconnect between uncertainty and volatility, and when 𝜂𝑡 is close to one disconnect is low. Rewriting Eq. (11) we have 𝜂𝑡 = 𝑡 𝑣𝑡 , (12) so high levels of 𝜂𝑡 capture situations in which the economy is in the ‘‘HL’’ regime, characterized by high uncertainty and low volatility, while low levels of 𝜂𝑡 are observed when the economy is in the ‘‘LH’’ regime, characterized by low uncertainty and high volatility. By contrast, in the two connected regimes ‘‘HH’’ and ‘‘LL’’, 𝜂𝑡 tends to be away from extreme values, and closer to its normalized mean value set to 1. Of course, the variables in Eq. (12) are continuous so the notion (and granularity) of any set of discrete regimes is simply a convenient low-dimensional representation of the investment environment that is useful for interpretation and aggregation, but plays no formal role in the analysis of the model. To fully specify the model, we assume that 𝜂𝑡 is a positive process that, like 𝑣𝑡 , is a general stochastic process with drift 𝜇𝑣,𝑡 and diffusion 𝜎𝑣,𝑡 under the reference probability measure.18 𝑑𝜂𝑡 = 𝜇𝜂,𝑡𝑑𝑡 + 𝜎𝜂,𝑡 (√ 1 − 𝜌 2 𝑆,𝜂𝑑𝑊 𝜂 𝑡 + 𝜌𝑆,𝜂𝑑𝑊 𝑆 𝑡 ) , 𝜂0 > 0. (13) The term 𝜌𝑆,𝜂 captures the correlation between the disconnect and stock price process. Furthermore, since volatility is correlated with the stock price, it follows that disconnect and the volatility are also correlated with correlation parameter equal to the product of 𝜌𝑆,𝜂 and 𝜌𝑆,𝑣. 4. Optimal portfolio allocation in partial equilibrium To solve the investor’s objective problem, we employ dynamic programming. We solve the resulting robust Hamilton–Jacobi– Bellman (HJB) equation under inequality constraints using the Lagrangian method. We then derive the optimal robust solution to the investor’s investment and consumption problem in closed form. Let 𝑋𝑡 denote the investor’s wealth and 𝜔𝑡 be the percentage of wealth (or portfolio weight) invested in the risky asset; with 1 − 𝜔𝑡 is invested in the risk-free asset. The investor consumes at an instantaneous rate 𝐶𝑡 and assumes the risky asset evolves according to the dynamics specified in Eq. (5). Accordingly, defining 𝜇 ℎ 𝑆,𝑡 ∶= 𝜇𝑆,𝑡 + 𝜎𝑆𝑣𝑡ℎ𝑡 , the dynamics of investor’s wealth 𝑋𝑡 follow 𝑑𝑋𝑡 = 𝜔𝑡𝑋𝑡 ( 𝑑𝑆𝑡 + 𝐷𝑡𝑑𝑡 𝑆𝑡 ) + ( 1 − 𝜔𝑡 ) 𝑋𝑡 𝑑𝐵𝑡 𝐵𝑡 − 𝐶𝑡𝑑𝑡 = ( 𝑋𝑡 [ 𝑟𝑓,𝑡 + 𝜔𝑡 ( 𝜇 ℎ 𝑆,𝑡 − 𝑟𝑓,𝑡)] − 𝐶𝑡 ) 𝑑𝑡 + 𝜔𝑡𝑋𝑡𝜎𝑆𝑣𝑡𝑑𝑊 𝑆,𝜗 𝑡 , (14) starting from an initial endowment 𝑋0 > 0. Under the robust probability measure P 𝜗 , the investor derives logarithmic utility from consumption with subjective discount rate 𝛽 > 0, and solves the infinite horizon problem19: sup {𝐶𝑠 ,𝜔𝑠 }𝑡≤𝑠<∞ inf {ℎ𝑠 }𝑡≤𝑠<∞ E 𝜗 𝑡 [ ∫ ∞ 𝑡 𝑒 −𝛽𝑠 log(𝐶𝑠 )𝑑𝑠] , (15) subject to the entropy growth constraint in Eq. (10) and the wealth dynamics in Eq. (14). In partial equilibrium, the investor solves this problem taking the dynamics of asset prices as given. The investor’s desire for robustness against model uncertainty is incorporated by evaluating the future evolution of the economy under the worst-case alternative model within the admissible set specified in Eq. (10). In order to solve the investor’s optimization problem, we make use of standard robust dynamic programming techniques. To this end, we define the value function 𝑉 (𝑡, 𝑋𝑡 , 𝑣𝑡 , 𝜂𝑡 ) = sup {𝐶𝑠 ,𝜔𝑠 }𝑡≤𝑠<∞ inf {ℎ𝑠 }𝑡≤𝑠<∞ E 𝜗 𝑡 [ ∫ ∞ 𝑡 𝑒 −𝛽𝑠 log(𝐶𝑠 )𝑑𝑠] , (16) 17 This approach to modeling a disconnect between uncertainty and volatility is not the only one possible. Other specifications, such as, for instance, an additive formulation can also be implemented and allow for explicit solutions within our framework provided that the investor has logarithmic utility. However, this simple multiplicative definition of uncertainty has two advantages: First, since disconnect is unobserved, it can only be identified once we have a proxy for volatility and uncertainty. By defining uncertainty 𝑡 as the product of both volatility and disconnect, we have a simple and a consistent (within our model) way of extracting an empirical measure for what we defined to be the disconnect process 𝜂𝑡 . Second, by imposing that 𝜂𝑡 is a strictly positive process, we do not have to be concerned with, for instance in the case that uncertainty decomposes additive into volatility and disconnect, how to interpret negative disconnect 𝜂𝑡 , and further technical issues as to whether the robust utility maximization problem is still well defined in the case when 𝜂𝑡 is allowed to change signs. 18 As was the case with volatility, the equilibrium is independent of the specific choice of the drift and diffusion for the disconnect process in the case of a logarithmic investor, who is myopic with respect to their specification. However, for the disconnect process to remain stationary and positive, just as in the case for the volatility process in Eq. (4), functional form and/or parameter restrictions on its drift and diffusive function are imposed. 19 Appendix A.4 derives the equilibrium when the investor has CRRA preferences using the martingale approach

Journal of Econometrics xxx (xxxx) xxx 8 Y. Aït-Sahalia et al. associated with the optimal stochastic robust control problem in Eq. (15). Then, as above, we define 𝜇 ℎ 𝑣,𝑡 ∶= 𝜇𝑣,𝑡 + 𝜌𝑆,𝑣𝜎𝑣,𝑡ℎ𝑡 and similarly 𝜇 ℎ 𝜂,𝑡 ∶= 𝜇𝜂,𝑡 + 𝜌𝑆,𝜂𝜎𝜂,𝑡ℎ𝑡 , the perturbed Hamilton–Jacobi–Bellman (HJB) equation characterizing the optimal solution is 0 = sup {𝐶𝑡 ,𝜔𝑡 } inf {ℎ𝑡 } { 𝑒 −𝛽𝑡 log(𝐶𝑡 ) + 𝜕𝑉 𝜕𝑡 + 𝜕𝑉 𝜕𝑋 ( 𝑋𝑡 [ 𝑟𝑓,𝑡 + 𝜔𝑡 ( 𝜇 ℎ 𝑆,𝑡 − 𝑟𝑓,𝑡)] − 𝐶𝑡 ) + 1 2 𝜕 2𝑉 𝜕𝑋2 𝜔 2 𝑡 𝑋 2 𝑡 𝜎 2 𝑆 𝑣 2 𝑡 + 𝜕𝑉 𝜕𝑣 𝜇 ℎ 𝑣,𝑡 + 1 2 𝜕 2𝑉 𝜕𝑣2 𝜎 2 𝑣,𝑡 + 𝜕𝑉 𝜕𝜂 𝜇 ℎ 𝜂,𝑡 + 1 2 𝜕 2𝑉 𝜕𝜂2 𝜎 2 𝜂,𝑡 (17) + 𝜕 2𝑉 𝜕𝑋𝜕𝑣 𝜔𝑡𝑋𝑡𝜎𝑆𝑣𝑡𝜌𝑆,𝑣𝜎𝑣,𝑡 + 𝜕 2𝑉 𝜕𝑋𝜕𝜂 𝜔𝑡𝑋𝑡𝜎𝑆𝑣𝑡𝜌𝑆,𝜂𝜎𝜂,𝑡 + 𝜕 2𝑉 𝜕𝜂𝜕𝑣 𝜌𝑆,𝜂𝜌𝑆,𝑣𝜎𝑣,𝑡𝜎𝜂,𝑡} , subject to the relative entropy growth constraint20 (𝜗𝑡 ) = ℎ 2 𝑡 2 ≤ 𝜖 2 2  2 𝑡 . (18) The robust optimal control problem is solved in two steps. First, the investor solves the inner optimization problem, deriving the optimal worst-case drift perturbation ℎ ∗ 𝑡 . Second, the investor solves the outer problem, selecting the optimal consumption and portfolio holdings that maximize his or her expected utility of consumption under the worst-case model. The solution for the optimal robust policies is characterized in the following proposition. Proposition 1. The optimal consumption, drift perturbation, and portfolio policy are given, respectively, by Consumption ∶ 𝐶 ∗ 𝑡 = 𝛽𝑋𝑡 . (19) Perturbation ∶ ℎ ∗ 𝑡 = −𝜖𝑡 = −𝜖𝜂𝑡𝑣𝑡 . (20) Portfolio weight ∶ 𝜔 ∗ 𝑡 = 𝜇𝑆,𝑡 − 𝑟𝑓,𝑡 𝜎 2 𝑆 𝑣 2 𝑡 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ reference demand − 𝜖 𝜎𝑆 𝜂𝑡 ⏟⏟⏟ . uncertainty correction (21) The investor’s optimal consumption is a constant fraction of wealth equal to the subjective discount rate 𝛽, which is a standard result with logarithmic utility. The optimal drift perturbation ℎ ∗ 𝑡 is negative, and driven by the product of uncertainty aversion and model uncertainty.21 Because model uncertainty is the product of stochastic disconnect and volatility, for a given level of uncertainty aversion, the drift adjustment can be high even when volatility is low, if there is high disconnect (corresponding to the HL regime). It is negative and increasing in magnitude in the investor’s degree of uncertainty aversion and in the level of disconnect. Finally, as Eq. (21) shows, the optimal portfolio holdings decompose into a standard myopic term equal to the Sharpe ratio, plus an uncertainty correction term. The correction term is largest when 𝜂𝑡 is largest, that is, in the HL regime where uncertainty is significantly higher than volatility, and lowest when 𝜂𝑡 is lowest, i.e., in the LH regime characterized by uncertainty being significantly lower than volatility.22 Conversely, in the case where 𝜂𝑡 is low, which implies high disconnect as volatility is high relative to uncertainty, the uncertainty correction term 𝜖𝜂𝑡∕𝜎𝑆 is small, but so is the portfolio holdings of the robust investor because the myopic term (𝜇𝑆,𝑡 − 𝑟𝑓,𝑡)∕𝜎 2 𝑆 𝑣 2 𝑡 shrinks as well. 5. Equilibrium asset prices Given the optimal demand for the assets expressed by the investor, we now solve for the equilibrium asset prices. An equilibrium is a specification of the dynamics of the risky and risk-less asset prices, including 𝜇𝑆,𝑡 = 𝜇𝑆 ( 𝑣𝑡 , 𝜂𝑡 ) and 𝑟𝑓,𝑡 = 𝑟𝑓 ( 𝑣𝑡 , 𝜂𝑡 ) , combined with a set of optimal robust consumption and investment policies that support continuous clearing in the markets for the consumption good and the risky asset. The two market clearing conditions are 𝐶𝑡 = 𝐷𝑡 , (22) 𝜔𝑡 = 1. (23) The next proposition characterizes the equilibrium risk-free rate 𝑟𝑓,𝑡 and the conditional equity premium under the reference measure, which can be expressed as 1 𝑑𝑡 E𝑡 [ 𝑑𝑆𝑡 + 𝐷𝑡𝑑𝑡 𝑆𝑡 ] − 𝑟𝑓,𝑡 = 𝜇𝑆,𝑡 − 𝑟𝑓,𝑡, (24) given the dynamics for the stock price in Eq. (3). 20 It can be shown that, under mild regularity assumptions, the solution we provide satisfies the transversality condition: lim𝑡→∞ E 𝜗 [ 𝑉 (𝑡, 𝑋𝑡 , 𝜂𝑡 , 𝑣𝑡 ) ] = 0. A formal proof of this sufficient condition is available from the authors upon request. 21 While in principle there are two roots for the candidate solution of ℎ ∗ 𝑡 = ±𝜖𝑡 , only the negative solution is consistent with the minimization in Eq. (15). Intuitively, the minimization means that the investor evaluates his or her policies under the worst-case alternative in the set of models under consideration. 22 Disconnect has two effects in the demand function for the risky asset in Eq. (21). First, in the partial equilibrium setting considered here, disconnect directly affects the optimal portfolio holdings of the robust investor. As Eq. (21) shows, the robust investor optimally reduces his or her exposure in the risky asset. Second, in general equilibrium, as we will later show, disconnect increases the risk premium, i.e. the equity premium appearing in the first term in the demand function will be itself a function of volatility and disconnect

Journal of Econometrics xxx (xxxx) xxx 9 Y. Aït-Sahalia et al. Proposition 2. In equilibrium, the price–dividend ratio is 𝑆𝑡∕𝐷𝑡 = 1∕𝛽. Using the optimal perturbation function ℎ ∗ 𝑡 in Eq. (20), the equilibrium equity premium under the reference measure P and the risk-free rate are given by Equity premium ∶ 𝜇𝑆,𝑡 − 𝑟𝑓,𝑡 = 𝜎 2 𝐷 𝑣 2 𝑡 + 𝜖𝜎𝐷𝜂𝑡𝑣 2 𝑡 . (25) Risk-free rate ∶ 𝑟𝑓,𝑡 = 𝛽 + 𝜇𝐷𝑣𝑡 − 𝜎 2 𝐷 𝑣 2 𝑡 − 𝜖𝜎𝐷𝜂𝑡𝑣 2 𝑡 . (26) As the results in Proposition 2 show, the equity premium and risk-free rate are time-varying and non-linear functions of volatility and disconnect. In particular, the uncertainty term embedded in disconnect generates a flight-to-quality-like correlation among asset returns. In periods of increasing uncertainty (increasing disconnect, for a given level of volatility) the demand for the risky asset decreases and the demand for the safe asset increases. Accordingly, in periods of increasing uncertainty the investor requires a higher equity premium to hold the risky asset, and is willing to accept a lower risk-free rate to hold the risk-less asset in equilibrium. The presence of stochastic volatility may amplify or diminish these effects depending on whether it is connected or disconnected from uncertainty, respectively. In the following sections we show that, because both volatility and disconnect jointly drive asset prices, our model is more consistent with the observed dynamics of asset prices, including during high-disconnect episodes. The structure of the equilibrium equity premium and risk-free rate can be also understood in the context of the optimal portfolio policy obtained in Proposition 1. The negative uncertainty correction implies that the representative investor is more conservative and prefers to hold less of the risky asset and more of the risk-free asset compared to a reference investor. In general equilibrium, the investor must allocate all his or her wealth to the risky asset (see Eq. (23)). Therefore, the last term in the equity premium boosts up the reward from holding the risky asset just enough so that the investor optimally chooses to allocate all his or her wealth to the risky asset. By the same token, the risk-free rate has to decrease in general equilibrium, so the investor optimally chooses not to hold the risk-free asset at all. Finally, it is likely that alternative approaches to model robustness (as discussed in footnote ) would also lead to an agent’s optimal portfolio holdings that are more conservative and command a higher risk premium (an additional positive ambiguity or uncertainty adjustment) and a lower equilibrium risk free rate. 6. Empirical analysis 6.1. Data The S&P 500 (logarithmic) returns including dividends obtained from the CRSP database serve as a proxy for the risky-asset return.23 The risk-free rate is the three-month (constant maturity) Treasury bill rate, which we obtain from the Fred St. Louis Database (Ticker ‘‘DGS3MO’’). Both series are available at the daily frequency from January 1, 1986 until December 31, 2020; we aggregate them to the weekly frequency. We construct our weekly measure of uncertainty 𝑡 using the daily news-based economic policy uncertainty (EPU) index developed by Baker et al. (2016).24 The daily EPU index is constructed by counting in the archives of well over one thousand US newspapers the number of articles that contain at least one term related to economic policy uncertainty from the following list:‘‘uncertain’’, ‘‘uncertainty’’, ‘‘economic’’, ‘‘economy’’, ‘‘Congress’’, ‘‘deficit’’, ‘‘Federal Reserve’’, ‘‘legislation’’, ‘‘regulation’’, and ‘‘White House’’. The daily EPU time series is volatile, and we smooth it by computing its moving average over the past week (5 trading days) to construct the uncertainty measure we use at the weekly frequency. The diffusive term of the stock price process is 𝜎𝑆,𝑡 ∶= 𝜎𝑆𝑣𝑡 where 𝜎𝑆 = 1, since we match 𝑣𝑡 to the square root of realized variance, which is computed daily from aggregated intraday returns sampled at five minutes frequency (78 observations per trading day). To obtain a weekly measure of realized variance, we sum the daily realized variance estimates. Based on the definition of disconnect in Eq. (12), we construct an empirical proxy for 𝜂𝑡 at the weekly frequency by computing the ratio of uncertainty as measured by EPU to realized volatility, and we then normalize this ratio to have an average value over the full sample equal to one. 6.2. Time-series properties of volatility and uncertainty In Fig. 2, we plot the time series of volatility and uncertainty, as well as the resulting disconnect process. From Panel A, we observe that uncertainty is substantially more volatile than volatility: the standard deviation of the EPU index is 50.1% compared to only 7.3% for realized volatility. While less volatile, realized volatility is more persistent than uncertainty: the first order autocorrelation of volatility is 0.87, compared to 0.74 for the EPU index. In Panel B, we plot the associated normalized disconnect time series. Since it is constructed as a normalized ratio of uncertainty and volatility, it is also volatile and persistent. The shaded areas in the charts mark the three high-disconnect periods described above: the financial crisis, the US 2016 election, and the Covid-19 pandemic. High disconnect occurs whenever 𝜂𝑡 is far away from 1. Consistent with Fig. 1, the correlation between volatility and uncertainty is only 0.44: while the two series co-move on average, they are often disconnected. Regressing uncertainty on volatility and a constant yields an adjusted 𝑅2 of 25%. Therefore, 75% of the time-series variation in uncertainty cannot be explained by volatility—the gap which we attribute to disconnect in the model. 23 Center for Research in Security Prices, US Stock Database, obtained from Wharton Research Data Services. 24 Section 6.5 below shows that our results continue to hold when considering alternative empirical measurements of uncertainty

Journal of Econometrics xxx (xxxx) xxx 10 Y. Aït-Sahalia et al. Fig. 2. Time series of volatility, uncertainty and disconnect. Notes: Panel A shows uncertainty (proxied by the economic policy uncertainty index EPU𝑡 ) and volatility (proxied by realized volatility). Panel B shows the disconnect process (𝜂𝑡 ) defined as the ratio of the EPU index divided by realized volatility, normalized to have a mean of 1. The sample period is from January 1986 until December 2020, and the data is sampled at the weekly frequency. The shaded areas correspond to five sub-periods associated with high disconnect: the mid 80’s (from January 1986 until November 1986), the early 90’s (from June 1991 until February 1996), the financial crisis (from July 2007 until March 2009), the US 2016 election (from July 2016 until January 2018), and the Covid-19 pandemic (from January 2020 until December 2020). Table 1 Average stock excess return and risk-free rate in selected high-disconnect regimes. Full Sample Mid 80’s Early 90’s Financial Crisis US 2016 election Covid-19 Time Period Begin Jan. 1986 Jan. 1986 Jun. 1991 Jul. 2007 Jul. 2016 Jan. 2020 Time Period End Dec. 2020 Nov. 1986 Feb. 1996 Mar. 2009 Jan. 2018 Dec. 2020 Stock excess return (%) 7.70 18.8 7.77 −33.40 17.29 16.97 Risk-free rate (%) 3.20 6.2 4.22 2.02 0.67 0.38 Uncertainty 𝑡 101.21 126.79 111.68 130.02 85.00 284.30 Stock return volatility 𝑣𝑡 (%) 17.70 9.78 7.32 28.16 10.51 28.94 Disconnect 𝜂𝑡 = 𝑡∕𝑣𝑡 (scaled) 1.00 1.52 1.77 0.59 1.29 1.49 Notes: This table presents the average annualized stock excess return, risk-free rate, uncertainty (𝑡 ), annualized volatility (𝑣𝑡 ), and disconnect (𝜂𝑡 ). These averages are computed over the full sample (from January 1986 until December 2020) and during five sub-periods associated with high disconnect: the mid 80’s (from January 1986 until November 1986), the early 90’s (from June 1991 until February 1996), the financial crisis (from July 2007 until March 2009), the US 2016 election (from July 2016 until January 2018), and the Covid-19 pandemic (from January 2020 until December 2020). All the data is nominal and sampled at the weekly frequency. 6.3. Average excess returns in high-disconnect periods Table 1 reports the average stock excess return and risk-free rate, along with the average uncertainty, volatility, and disconnect for the full sample and for the following high-disconnect sub-periods: the mid 80’s (from January 1986 until November 1986), the early 90’s (from June 1991 until February 1996), the financial crisis (from July 2007 until March 2009), the US 2016 election (from July 2016 until January 2018), and the Covid-19 pandemic (from January 2020 until December 2020). Over the full sample, the average stock excess return is 7.7%, with a relatively high level of the average risk-free rate compared to recent values. Given the paths of uncertainty 𝑡 and volatility 𝑣𝑡 , the normalized measure of disconnect 𝜂𝑡 has an average value of 1 over the full sample by construction. Before analyzing the most recent disconnect sub-periods, over which we will examine in more detail the performance of the model, we briefly describe two older high-disconnect periods: the mid 80’s and the early 90’s. Compared to the full sample results, both of these periods are characterized by a very high risk-free rate (about double its level in the full sample for the mid 80’s and